s- and p-polarization

When light strikes a surface at an oblique angle, the incident ray and the surface normal define the plane of incidence. Any polarization state can be split into two linear components relative to that plane: s-polarized light has its electric field perpendicular to it, and so parallel to the surface, and p-polarized light has its electric field in the plane of incidence. The letters come from the German senkrecht (perpendicular) and parallel. The two components obey different boundary conditions, so they reflect and transmit by different amounts. At 45° on uncoated glass of index 1.5 the s reflectance is 9.2% and the p reflectance 0.85%, a ratio of about 11; at normal incidence the plane of incidence is undefined and both reflect 4.0%.

Reflectance of the two polarizations

The Fresnel equations give the amplitude reflection coefficients for light going from index $n_1$ to $n_2$ at angle of incidence $\theta_1$ and refraction angle $\theta_2$:

$$r_s = \frac{n_1\cos\theta_1 - n_2\cos\theta_2}{n_1\cos\theta_1 + n_2\cos\theta_2},$$ $$r_p = \frac{n_2\cos\theta_1 - n_1\cos\theta_2}{n_2\cos\theta_1 + n_1\cos\theta_2},$$

with reflectances $R_s = |r_s|^2$ and $R_p = |r_p|^2$. Computed for air to glass of index 1.5:

$R_s$ rises steadily toward 100% at grazing incidence. $R_p$ first falls to zero at the Brewster angle, $\theta_B = \arctan(n_2/n_1)$, which is 56.3° for $n$ = 1.5, and then rises.

Why 45° optics are polarization-sensitive

Fold mirrors, beam splitters and dichroic mirrors are usually used at 45°, where s and p behave differently at every interface of the coating. An uncoated glass plate at 45° transmits 82.4% of s-polarized light and 98.3% of p-polarized light through its two surfaces, ignoring multiple reflections, so even a window tilted in a beam path changes the polarization state of light that is not purely s or p. In a dielectric coating the effective layer indices differ for the two polarizations, so the reflection band of a 45° mirror is narrower for p than for s, and the cut-on edge of a 45° dichroic sits at different wavelengths for the two, often a few to tens of nanometers apart, by an amount that depends on the design. A polarizing beamsplitter exploits the difference: its coating reflects s strongly and transmits p, so it also serves as a polarizer.

Metal mirrors at 45° also introduce a phase difference between s and p on reflection, so linearly polarized light at an intermediate azimuth comes back elliptically polarized. Keeping the input polarization purely s or purely p with respect to each fold mirror avoids this.

Brewster windows and Brewster plates

Light that meets a dielectric surface at the Brewster angle loses no p-polarized power to reflection. Gas lasers such as the helium-neon laser close their tubes with windows tilted to this angle; inside a window of index 1.5 the beam meets the second surface at 33.7°, which is the internal Brewster angle, so p light passes both surfaces without loss. The s component loses 14.8% at each surface, 27% per pass through the window and about 47% per round trip through one window, so only p polarization reaches threshold and the output is linearly polarized.

s and p versus TE and TM

At a single interface, s-polarized light is called transverse electric (TE) and p-polarized light transverse magnetic (TM), and in a slab waveguide the TE modes are built from s-polarized rays bouncing between the layers (TE and TM polarization). The labels can still clash in practice. On a photonic chip or an edge-emitting laser, "TE" means an electric field parallel to the chip surface, a property fixed relative to the device. Once that beam leaves the chip and meets a free-space mirror, whether it is s or p depends only on how the mirror is tilted relative to the field: a TE beam from a laser lying flat on the table is s-polarized on a mirror that folds the beam vertically and p-polarized on one that folds it horizontally. In the thin-film and ellipsometry literature, TE and TM are used as synonyms for s and p. Older texts that named the plane of polarization after the magnetic field describe s light as polarized in the plane of incidence, so the definition is checked against the direction of the electric field.

Common questions

Which reflects more, s or p polarization?

s-polarized light reflects more strongly from a dielectric surface at every angle between normal and grazing incidence, or up to the critical angle for internal reflection. At normal incidence the two are equal.

Is s-polarization horizontal or vertical?

Neither in general; it depends on the plane of incidence. For a mirror on a table that folds the beam horizontally, the plane of incidence is horizontal, so s is vertical and p is horizontal. Glare from water is mostly s-polarized, which is horizontal in the room.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017); M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); H. A. Macleod, Thin-Film Optical Filters, 4th ed. (CRC Press, 2010).